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Tuesday, October 20, 2020

**Abstract:** The arithmetic properties of the ordinary partition function p(n) have been the topic of intensive study. Ramanujan proved that there are linear congruences of the form p(ℓn+β)≡0(modℓ) for the primes ℓ=5,7,11, and it is known that there are no others of this form. On the other hand, there are many examples of congruences of the form p(ℓQmn+β)≡0(modℓ) where Q is prime and m≥3. Here we prove that such congruences are very scarce in the case when m=1 or m=2. The proofs rely on a variety of tools from the theory of modular forms and from analytic number theory. This is joint work with Olivia Beckwith and Martin Raum. Please contact Kevin Ford (ford@math.uiuc.edu) for Zoom link.